Circadian pattern and burstiness in human communication activity

Related work

• Prior work attributed approximate power-law inter-event distributions to circadian and weekly inactivity through cascading inhomogeneous Poisson processes.These models combine time-dependent activity rates with shorter-timescale cascading burst behavior.Source: §I · p. 1
• Other studies suggested correlations beyond time cycles and modeled bimodal Short Message intervals without incorporating circadian or weekly patterns.The omission of seasonal effects left their contribution to burstiness unresolved.Source: §I · p. 1

Motivation

• De-seasoning tests whether heavy-tailed communication intervals persist after circadian and weekly activity patterns are removed.The question matters for distinguishing inhomogeneous Poissonian explanations from models incorporating intrinsic correlations.Source: §I · p. 1

Approach

• The paper extracts seasonal patterns, rescales event timings to remove them, and compares rescaled and original inter-event distributions.Time is dilated during high activity and contracted during low activity.Source: §I · pp. 1, 3

Contribution

• The study concludes that heavy tails and burstiness are mostly due to intrinsic correlations rather than circadian and weekly cycles.The de-seasoning procedure also improves the distributions' scaling behavior.Source: §I · p. 3

Data

• The datasets contain 119 days of bidirectional mobile-call and Short Message records from a European operator, with 322 × 10^6 calls and 114 × 10^6 messages.The records include 5.2 × 10^6 mobile-call users and 4.2 × 10^6 Short Message users.Source: §II · p. 3

Homogeneous rescaling

• Event density c(t) provides the observable for circadian and weekly cycles, while rescaled time is constructed from the average density over the 119-day period.The transformation is based on the cumulative activity rate.Source: §II · p. 3
• Rescaling transforms event rates so that ρ*(t*) = 1, dilating time during high activity and contracting it during low activity.In the rescaled frame, the cyclic pattern is removed by construction.Source: §II · p. 3

Activity-dependent rescaling

• Homogeneous rescaling uses one transformed timeline for all nodes, whereas activity-dependent rescaling assigns separate event-rate transformations to node or link activity classes.The class-specific rates differ across activity levels.Source: §II · p. 3

Mobile phone calls

• Mobile-call activity has inactive nights, active daytime, separate weekday and weekend profiles, and the week's largest peak on Friday evening.Weekend peaks are more equal, while weekday afternoon peaks are smaller than evening peaks.Source: A. Homogeneous rescaling · p. 3

De-seasoning outcome

• The mobile-call rescaled event rate reaches ρ*(t*) = 1 with only weak fluctuation, measured as σ ≈ 0.012.The remaining fluctuations retain circadian and weekday-versus-weekend structure, indicating heterogeneous node and link activity.Source: A. Homogeneous rescaling · p. 3

Short Message patterns

• σ ≈ 0.011 is the overall deviation after homogeneous rescaling of Short Message activity.Weekdays show a distinct midnight peak that is absent on weekends.Source: A. Homogeneous rescaling · p. 3

Classification and rescaling

• Node- and link-activity classifications divide entities into ten activity classes, with separate event rates and rescaling for each class.Node classes are defined by node event counts, while link classes are defined by event counts between node pairs; the two schemes cannot be used simultaneously.Source: B. Activity dependent rescaling · pp. 3–4

Class-specific temporal structure

• Class-specific event rates retain the common inactive-nighttime and active-daytime cycle but exhibit different higher-order structures across activity levels.For MPC, the least active node class is generally below the homogeneous weekly pattern, whereas the most active class shows especially strong nighttime activity; link classes are qualitatively similar with differences among the most inactive classes.Source: B. Activity dependent rescaling · pp. 4–5, Figure 3 · p. 5

Activity-dependent de-seasoning

• 4.9 × 10−3 for node classes and 5.4 × 10−3 for link classes, versus 0.012 for homogeneous rescaling, showing improved de-seasoning of MPC activity.The rescaled event-rate fluctuations become closer to white noise when higher-order activity structures are included.Source: B. Activity dependent rescaling · pp. 6–7
• 6.0 × 10−3 for SM node classes and 6.9 × 10−3 for link classes after activity-dependent rescaling, with rescaled rates fluctuating around 1.The SM activity classes provide a more effective decomposition of aggregated activity than the MPC classes.Source: Figure 4 · p. 6, B. Activity dependent rescaling · p. 7

Class-specific temporal structure

• The SM classes show distinct temporal profiles, including weak nighttime activity for the least active node class and stronger nighttime activity for the most active classes.Similar behavior appears for link classes, and the aggregated SM activity is decomposed into classes better than MPC activity.Source: Figure 4 · p. 6, B. Activity dependent rescaling · p. 7

Spectral validation

• Circadian and weekly peaks disappear from the rescaled power spectra for both MPC and SM time series.The comparison uses power spectra of original and rescaled event rates to verify removal of the seasonal peaks.Source: B. Activity dependent rescaling · p. 7, Figure 5 · p. 7

Overall comparison

• Node- and link-activity-dependent rescalings improve de-seasoning by a large amount over homogeneous rescaling.This conclusion follows from the weaker rescaled-rate fluctuations in both MPC and SM analyses.Source: B. Activity dependent rescaling · p. 7

III. BURSTINESS

• Activity-dependent rescaling uses class-specific event rates, while links joining different node classes receive the average of their nodes’ rescaled times.The averaging choice resolves the non-unique rescaled time for links whose endpoints belong to different activity classes; analogous averaging is used for nodes in the link-class scheme.Source: §III · pp. 7–8
• The study treats boundary inter-event times differently for nodes and links to avoid interpreting a first link contact as a preceding waiting period.For links, only the interval after the last observed event is counted, whereas node intervals can include time before the first event.Source: §III · p. 8
• Rescaled distributions suppress the original 16-hour hump and circadian peaks, although peaks near daily multiples change little.The link-class rescaling method shows the best overall performance, and suppressed weekly peaks move toward smaller τ values.Source: §III · pp. 8–9, Figure 6 · p. 8, Figure 7 · p. 9
• The de-seasoning procedure weakly affects node and link burstiness, indicating that burstiness is not removed by seasonal adjustment.Burstiness is computed from the inter-event-time distributions after rescaling and compared with the originals.Source: Figure 8 · p. 9, §III · p. 10
• Heavy-tail scaling persists after de-seasoning, with improved scaling quality because circadian peaks are suppressed.For MPC, the first regime P(τ) ∼τ^-α has α ≈0.7 and extends to larger τ; SM shows a regime with α ≈1 extended to about 1.5 decades.Source: Figure 6 · p. 8, Figure 7 · p. 9, §III · p. 10
• The analysis cannot conclusively distinguish a second tail-scaling regime from an exponential cutoff.This uncertainty applies to the large-τ tails in both MPC and SM distributions.Source: §III · p. 10

• The study also classifies nodes and links by activity to examine higher-order structures in human communication.Source: §IV · p. 10
• Heavy tails and burstiness persist after de-seasoning, indicating they are not simply consequences of circadian and weekly cycles but reflect intrinsic correlation-based inhomogeneity.Source: §IV · p. 10
• The de-seasoning procedure improves the inter-event-time distribution’s scaling behavior by suppressing circadian peaks in the tails.Source: §IV · p. 10